chapter 3

# chapter 3 - 3 a x2 10 Optimal Solution(3,7 Line A x1 3x2 =...

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3. a. 8 4 4 8 x 2 x 1 2 6 10 2 6 0 10 Line B x 1 + x 2 = 10 Optimal Solution (3,7) Line A x 1 + 3 x 2 = 24 Feasible Region Optimal Value = 27 b. Slope of Line B = -1 Slope of Line A = -1/3 Let C 1 = objective function coefficient of x 1 C 2 = objective function coefficient of x 2 -1 - C 1 /3 -1/3 1 C 1 /3 C 1 /3 1/3 C 1 3 C 1 1 Range: 1 C 1 3 c. -1 -2/ C 2 -1/3 1 2/ C 2 2/ C 2 1/3 C 2 2 C 2 6 Range : 2 C 2 6 d. Since this change leaves C 1 in its range of optimality, the same solution ( x 1 = 3, x 2 = 7) is optimal. e. This change moves C 2 outside its range of optimality. The new optimal solution is shown below.

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8 4 4 8 x 2 x 1 2 6 10 2 6 0 10 Feasible Region 6 1 2 3 4 5 11. a. Regular Glove = 500 Catcher’s Mitt = 150 Value = 3700 b. The finishing and packaging and shipping constraints are binding. c. Cutting and Sewing = 0 Finishing = 3 Packaging and Shipping = 28 Additional finishing time is worth \$3 per unit and additional packaging and shipping time is worth \$28 per unit. d. In the packaging and shipping department. Each additional hour is worth \$28. 18. a. E = 80, S = 120, D = 0 Profit = \$16,440 b. Fan motors and cooling coils c. Labor hours; 320 hours available. d. Objective function coefficient range of optimality No lower limit to 159. Since \$150 is in this range, the optimal solution would not change. 20. a. Manufacture 100 cases of model A Manufacture 60 cases of model B Purchase 90 cases of model B Total Cost = \$2170 b. Demand for model A Demand for model B Assembly time
c. Constraint Dual Price 1 -12.25 2 -9.0 3 0 4 .375 If demand for model A increases by 1 unit, total cost will increase by \$12.25 If demand for model B increases by 1 unit, total cost will increase by \$9.00 If an additional minute of assembly time is available, total cost will decrease by \$.375 d. The assembly time constraint. Each additional minute of assembly time will decrease costs by \$.375. Note that this will be true up to a value of 1133.33 hours. Some students may say that the demand constraint for model A should be selected because decreasing the demand by one unit will decrease cost by \$12.25. But, carrying this argument to the extreme would argue for a demand of 0. 25. a. Let P 1 = units of product 1 P 2 = units of product 2 P 3 = units of product 3 Max 30 P 1 + 50 P 2 + 20 P 3 s.t. 0.5 P 1 + 2 P 2 + 0.75 P 3 40 Machine 1 P 1 + P 2 + 0.5 P 3 40 Machine 2 2 P 1 + 5 P 2 + 2 P 3 100 Labor 0.5 P 1 - 0.5 P 2 - 0.5 P 3 0 Max P 1 -0.2 P 1 - 0.2 P 2 + 0.8 P 3 0 Min P 3 P 1 , P 2 , P 3 0 A portion of the optimal solution obtained using The Management Scientist is shown. Objective Function Value =

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chapter 3 - 3 a x2 10 Optimal Solution(3,7 Line A x1 3x2 =...

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